INTRODUCTION

Mathematico-Physics of Generalized Calculus

Module-0

Shantanu DasRRPS

Reactor Control DivisionBARC

2009-2010

Exact thinking?

Sir Isaac Newton & Gottfried Wilhelm Leibniz independently discovered Calculus in the middle of 17th century. In recognition to this remarkable discovery, J. Von Neumann remarked ,

“……the calculus was the first achievement of modern mathematics and it is difficult to overestimate its importance. I think it defines more equivocally than anything else the inception of modern mathematical analysis, which is logical development, still constitutes greatest technical advance in exact thinking”

Is that all in exact thinking?

Why this subject?

Myself neither “ a mathematician nor a physicist” yet I picked up this subject about more than a decade ago, to have perhaps exact thinking, for making:

Fuel Efficient Nuclear Power Plant Controls

I discovered that this subject as:

“…..the language what our nature understands the best; and could be a efficient way to have better communication with nature.”

A language of efficient communication with the process makes myself to have efficient controls

The Greatest Discovery

One of the greatest discovery that Newton made was that:

“The nature follows Mathematics”

The intriguing question for all of us even today is

Which Mathematics?

That is why we are here discussing this Mathematico-Physics of Generalized Calculus

Representing real life systems as “Non-Linear” dynamics systems, “chaotic” systems, “irregular” systems is of increasingly of interest in interdisciplinary subjects; engineering, physical-chemical science, biological science, economics, medicine, geo-physics and so on.

The ‘non-integer’ representation of dimensions or ‘fractal’ dimensions is one possible parameter to characterize a chaotic, non-linear, continuous but non-differentiable systems.

A non-differential system, irregular system, erratic system, chaotic system, rough system, noisy system, has measure parameter as fractal dimension and can be ‘fractionally differentiated’ (otherwise difficult).

It is about fractional integration fractional differentiation and fractal dimensions all are generalizations of normal integer order calculus and normal Euclidian dimensions, which enable us to possibly extract information out of these irregular behavior.

Well nature is non-linear complex fractal irregular not well behaved, far too complex to have our exact thinking to understand, well Fractional Calculus a fractional step to understand ournature.

A passing mention to Non-Linear-Irregular Dynamics

Salute to Indian Mathematicians of Fractional Calculus

V. BalakrishnanAnil.D. Gangal

Kiran .M. KolwankarH.M.Srivastava

O.P.AgarwalS.C Dutta Ray

L.DebnathR.K.SaxenaR.K.Raina

Rasajit Kumar Bera…………………………

and to all exponents around the globe to have given this wonderful subject to us applied scientists and engineers, a language what nature understands the best, and thus to

communicate with the nature in better and efficient way.

Implementation of fractional calculus is…….

………….in understanding nature better

………….in making effort to have this subject as “Popular Science”

………….in simple teaching and evolving the simple methods in mathematics

…………..in making working systems

……………in realizing that our physical understanding is limited and mathematical tools go far beyond our understanding

……………in appreciating the wonderful world of mathematics that lays between integer order differentiation and integration

…………….in appreciating & finding new physical meanings

Birth of Fractional Calculus

The concept of differential operator D=d/dx is familiar to all who studied elementary calculus (Newtonian-Leibniz integer order calculus). For a suitable function f , the n-th derivativenamely is well defined.

In 1695, September 30th L’ Hopital inquired Leibniz, what could be ascribed to If ‘n’ be were fraction (say ½).Leibniz responded as apparent paradox today, will result in useful consequences tomorrow.

On that day Fractional Calculus was born

Since that time the ‘fractional calculus’ has drawn attention of many.Euler, Laplace, Fourier, Abel, Riemann, Liouvelli, Laurent,……

This is three hundred year old mathematical tool, perhaps will find its application for applied science & engineering in XXI century.

( ) ( ) /n n nD f x d f x d x≡

nD f

Fractional Calculus does not mean the calculus of fractions, nor does it mean a fraction of any calculus, differentiation, integration or calculus of variations.

The FRACTIONAL CALCULUS is a name of theory of integration and derivatives of arbitrary order, which unify and generalize the notion of integer order n-fold repeated differentiation and n-fold repeated integration.

FRACTIONAL CALCULUS is GENERALIZED differentiation and integration.

GENERALIZED DIFFERINTEGRATIONS

What is not FRACTIONAL CALCULUS

The Generalized Calculus

( )

,

i

i

d f zd z

e

α β

α β

α β

+

+

∈ ℜ

( )d f td t

e

α

α

α ∈ ℜ

( )n

n

d f xd x

n Z∈

Integer OrderNewtonianPoint propertySymmetry

Fractional order

Non-localDistributedHistory/heredityNon-MarkovianIrreversibly

Complex-Order

Imaginary time responsealong with real time, fractal oscillatory frequency response

Generalization of theory of numbers and calculations

32 2 2 2 8= × × = Can be visualized0 .52 ex p { (0 .5) ln 2} 1 .41 4= = Number exists but hard to visualize how.

5 ! 1 2 3 4 5 120= × × × × = Is a visualized quantity, but what about (5.5)!

Generalized factorial as GAMMA FUNCTION (5.5)! (1 5.5) (6.5) 287.88= Γ + = Γ =

ln

1

0

,! ( 1) ( )

( )

( !)( ) l im( 1) ( 2 ) . . . ( )

r r x

t x

x

n

x e r ex x x x

x e t d t

n nxx x x x n

∞− −

→ ∞

= ∈ ℜ= Γ + = Γ

Γ =

Γ =+ + +

∫

fdfdx

2

2

d fdx

3

3

d fdx

f d x∫fd x∫∫

Wonderful universe of mathematics lays in between one full integration and one full differentiation

Fractional calculus gives continuum between full differ-integration

2

/

/ / 2 2

( )( ) ( ) / 2( ) ( ) / 2

f x xf x d f x d x xf x d f x d x

=

= =

= = ( )

0 1

d f xd x

α

α

α≤ ≤

0 2( ) ( )f x D f x x= =/ 1( ) ( ) 2f x D f x x= =

/ 1( ) ( ) 2f x D f x x= =

// 2( ) ( ) 2f x D f x= =

1 .5 ( )D f x

Curve fitting will be effective by use of fractional differential equation, as compared with polynomial regression and integer order differential equation. The reason is extra freedom to closely track the curvature in continuum.Could be a magnifier tool to observe the formation of discontinuity.

Fractional derivative the Euler (1730) formula for monomial

{ }

{ }

{ }0 .5

1 0 .50 .5

( ) .................... ( )

( 1)( 2 )............( 1)

( 1) ( 1)( 2 )........( 1) ( 1)( 1)

( 1)

(1 1)(1 0 .5 1) (1

n

n

nn

m m nn

nm m n

n

d f x d d d f xd x d x d x d x

d x m m m m n xd x

m m m m m n m nd mx xd x m n

d xx xd x

−

−

−

=

= − − − +

Γ + = − − − + Γ − +

Γ +=Γ − +

Γ += =Γ − + Γ

20 .5) 0 .5 (0 .5)

x xπ

= =+ Γ

For positive index the process is differentiation For negative index the process is integration

Discrete difference to continuum limitBackward difference (construction of up-shift operator)Backward difference of Stochastic process & Taylor’s expansion

22( ) ( ) ( ) ( ) . . . ( ) . . . [ ] ( )

2 ! !

nn DX t X t D X t D X t D X t e X t

nττ ττ τ+ = + + + + + =

Derivative operator can define ‘shift operator asD { }( ) ( )E E X t X tτ τ τ≡ = +

From above Taylor's expansion we can write:Also we can formulate:

{ }( ) ( )DE X t e X tττ =

( )1 DE e ττ −≡ − Δ =

Backward difference operator or Up-shift operator is:2

( )( )1 1 1 1 ...

2D DE e D Dτ

τττ τ−

⎛ ⎞Δ = − = − = − + + + ≈ −⎜ ⎟

⎝ ⎠

In a way we have defined a backward derivative operator as: ( )

0l i m Dτ τ

−

→

Δ⎛ ⎞= −⎜ ⎟

⎝ ⎠

It is suggestive that is time derivative operator and when applied to a continuous function yields: ( )

0

( )( ) l i m

X td X td t τ τ

−

→

Δ= −

D

Forward Difference (Construction of down-shift operator)

1 ( ) ( )E X t X tτ τ− ≡ −

From corresponding Taylor’s expansion we can get: 1( )1 DE e τ

τ− −

+≡ − Δ =

Therefore 1( ) 1 1 DE e τ

τ− −

+Δ = − = −

and a forward derivative operator.( )

0l im Dτ τ

+

→

Δ⎛ ⎞=⎜ ⎟

⎝ ⎠

Backward derivative operator and Forward derivative operator are equivalent mathematical expression of the Derivative. To be on safe side a physicist or engineer often writes the derivative operator in symmetric way as:

( ) ( )

0l i m

2dDd tτ τ

+ −

→

Δ − Δ= =

With similar explanation as previous slide

Stochastic difference equation

Random time series are traditionally modeled in physical science using ‘randomwalks’ . If is ‘random variable’ intending to represent a step taken at discrete time then the random walk variable that denotes the total distancetraveled in time or after such steps is:

jξj

t N τ= N1

( )N

jj

X t ξ=

= ∑

Alternatively ( ) ( ) NX t X t τ ξ− − =

We can simplify the notation by setting and with down shift operator in unit interval without subscript as:

Above is discrete analog of Brownian Motion and is Physicist’s guide into the world of Stochastic Process

1τ =1(1 ) j jE X ξ−− =

Can we generalize and obtain a Fractional Stochastic Process, making Fractional Brownian Motion, with non-integer like the following?

1( 1 ) j jE Xα ξ−− =α

And why not?

Mean Squared Displacement- a revisit:

MSD is vector distance traveled by ‘walker’ over someinterval of length time ; its magnitude squared is averaged over many suchintervals . If no other ‘walker’ is encountered the walker traveling balasticallythen the distance traveled would be proportional to the time ( times with velocity) and MSD would increase quadratic ally as

In denser phases the time square behavior holds only for short time of the order of collision time. Beyond that time motion be better described as ‘random-walk’, for which MSD is proportion to time , that is

The rate of growth of MSD depends on how often the walker suffers collision. At higher density it will take longer to diffuse a given distance as other walkers willimpede its progress. The limiting slope of MSD considered for sufficiently longtime interval for it to be in linear region is related as (by Einstein) as Diffusionconstant and space dimension d.

Well constant is function; dimension d can be non-Euclidian a fraction!!

22 2

0

1( ) ( ) ( ( ) ( 0 ) )N

j j j jj

X t X t X t XN =

= Δ = −∑

( ) (0)j jX t X−t

t( )jX t vt=

2 2jX t≈

2t

2jX t≈

2l i m 2t

d X dd t→ ∞

= D

Non-Linear MSD cases-Anomalous diffusion

2 2( ) HX t t≈

12H = Normal Random walk and normal diffusion

Brownian Motion, with no memory

2 ( )X t t≈

10 2H≤ < Anti Persistent Random walk and sub (slow) diffusionWith short term memoryThe process decays monotonically to zerohyperbolically

12 2( )

1 / 4

X t t

H

≈

=

11 2H> > Persistent Random walk and super (fast) diffusionLong-term- ‘lingering' memory (Long Ranged Dependence) LRD, Autocorrelation decays as power law, Fractional Brownian Motion

32 2( )

3 / 4

X t t

H

≈

=

H Hurst exponent

MSD from Fick’s diffusion and Brownian Motion of Perin:

The Fick’s law of diffusion is:

2

2

( , ) ( , )N x t N x tAt x

∂ ∂=

∂ ∂

Taking Fourier-Laplace transform of this we get:

02

( )( , ) N kN k ss A k

=+

2

20

( , ) ( ) ( , )

( , ) ( , 0 ) ( , )

N k t A ik N k tt

s N k s N k A k N k s

∂=

∂+ = −

t

x

si k

∂ →∂ →

The MSD is calculated from as (moment calculations):( , )N k s2

2 1 12 20

2l im ( , ) 2k

d Ax N k s A td k s

− −

→

⎧ ⎫⎡ ⎤⎪ ⎪ ⎧ ⎫Δ = − = =⎨ ⎬ ⎨ ⎬⎢ ⎥ ⎩ ⎭⎪ ⎪⎣ ⎦⎩ ⎭L L

The above takes for 0 ( ) 1N k = ( , 0 ) ( )N x xδ=

Maxwell-Debye exponential relaxation process:

0

0

1

0 1

( ) ( )

0 ; ( 0 )

( )

1( ) ( )

t

d t td t

t

t e

dt td t

τ

τ

τ

−

−

−

Φ = − Φ

> Φ = Φ

Φ = Φ

Φ − Φ = − Φ

Standard Maxwell Debye relaxation is

Gives pure exponential solution withsingle relaxation time constantAlso the Integral representation of Maxwell-Debye relaxation is:

The integral equation can be formally extended to Fractional Integral equation byreplacing which leads to1

1

1 1d dd t d t

β

β βτ τ

− −

− −→

0 01( ) ( )tt D tββτ

−Φ − Φ = − Φ

1

0 1

1( ) ( )dt td tτ

−

−Φ − Φ = − Φ

Non Debye non-exponential relaxation process:

Kohlraush Williams Watts (KWW) relaxation law:

( )0( )

t

t eα

τ⎧ ⎫−⎨ ⎬⎩ ⎭Φ = Φ

Nutting Power Law relaxation:

( )0( ) 1 ; 0 1ntt nτ

−Φ = Φ + < <

Observed in:

Dielectric relaxation, Stress Relaxation, Strain relaxation, NMR relaxationDiffusion controlled relaxation, electrical circuits,…….; unlike normal relaxation 0( )

tt e τ−Φ = Φ

Memory and Fractional Calculus:

1. Non-exponential relaxation implies MEMORY i.e. the underlying fundamental relaxation process are NON-MARKOVIAN

2. Natural way to incorporate memory effect is fractional calculus via the involved convolutionintegral in time (space). The present state is being influenced by all the states, the system has been running through at the times

3. The power-law Kernel defines the fractional expression represents a particular long memory.

0 , 1 , . . . . . . . . . .t t′ =

1

1

1( ) ( ) ( ) ( ) * ( )( )

1( ) ( ) ( )( 1) !

t

c tc

xn n

c xc

D f t t f d t f t

D f x x f dn

β ββτ τ τ

β

ξ ξ ξ

− −

− −

= − = ΡΓ

= −−

∫

∫

Memory Integrals:

0

( ) ( ) ( ) ( ) * ( )td t K t d K t t

d tτ τ τΦ

= − − Φ = − Φ∫

Represents Memory Integral i.e. all instances for to contribute to situationat

0τ = tτ =tτ =

1. Memory breaks down i.e. Markovian Case:

{ }

0

0 00

0 0

( ) ( )

( ) ( ) ( ) ( )

( ) e x p

t

K t K t

d t K t d K td t

t K t

δ

δ τ τ τ

=

Φ = − − Φ = − Φ

Φ = Φ −

∫

2. The opposite case Constant Memory i.e. leading to oscillatory case

02

02

0 0

( )

( ) ( )

( ) c o s ( )

K t K

d t K td t

t K t

=

Φ = − Φ

Φ = Φ

How the process time constant is related to Kernel of Memory integral, good research case?

( ) ( ) ( ) * ( )d t t K t td t τΦ Φ

= − = − Φ

3. Slowly varying Kernel which for small time behaves as power law gives KWWrelaxation process

0( )K t K t γ≈

{ }20 0( ) e x pt K t γ +Φ = Φ −

4. Relaxation for Fractional Differential/Integral equation & its Memory Kernel

[ ]

20

10

10

( ) ; 0 21( ) ( )

( 1 )

q

qtq

q

K t K t qd t D td t

K qτ

τ

−

−

−

= < ≤

⎡ ⎤Φ = − Φ⎣ ⎦

= Γ −

Apply on both sides to get:10 tD −

0 0( ) ( )q qtt D tτ − −Φ − Φ = − Φ

Apply on both sides to get FDE 0

qtD

0 0( ) ( )(1 )

qq q

ttD t t

qτ

−−Φ − Φ = − Φ

Γ −

Memory Integral (Contd.)

Using Fractional Derivative of constant C as , non zero, that is C / (1 )qt q− Γ −

0

( )tΦΦ

1

0Time

02 , ( )q K t K= =

01, ( ) ( )q K t K tδ= =

1 .5q =

0 .5q =

0 0( ) ( )( 1 )

qq q

ttD t t

qτ

−−Φ − Φ = − Φ

Γ −

Memory Kernel & Fractional Differential Equation for Relaxation kinetics

MARKOVIAN PROCESS:

Modeling by Markovian way, assumes that the process during its functional activity assumes a number of conformal states. Transition between the statesare usually formulated as kinetic or relaxation rate constants (time constants)

- indicate how many conformational states of energylandscapes are taking part in relaxation process. Markovian models assumethat probability distribution function for some events may be described by sum of exponential functions.

The amplitude and are presumed to be independent and not related. Ifthey are correlated the Markovian is lost.

1 ; ( 1, 2 ... )n nk n Nτ −= =

0 10 1

00 1

( ) . . . . . . N N

Nt t ttN n

NnN n

a a aaf t e e e eτ τ ττ

τ τ τ τ− − −−

=

= + + + = ∑

naNτ

1 22

2

; 1

; 0 1

( ) . . . . . .

nn

nn N

t t tN

N

a a p p

a p pf t e e eλ λ

τ τ τ

τ τ λ λ

τ λ λ

− −⎛ ⎞ ⎛ ⎞ ⎛ ⎞− − −⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

= >

= < <

⎡ ⎤= + + +⎢ ⎥

⎢ ⎥⎣ ⎦

Fokker Plank Kolmogorov Equation

Basis from random walker theory and with MARKOVIAN model we get, theevolution of probability density as:

[ ] [ ]2

2( , ) ( ) ( , ) ( ) ( , )P x t A x P x t B x P x tt x x∂ ∂ ∂

= +∂ ∂ ∂

( )A x is the drift (Advection) ( )B x is the diffusion function

For a case of no advection, for force/potential free case and with constant B(x)=B we have for initial delta function

A Gaussian evolution

( , 0 ) ( )P x xδ=2

41( , )4

xtP x t e

tπ

⎛ ⎞− ⎜ ⎟⎜ ⎟Β⎝ ⎠=

Β

A Fractional Fokker Plank Kolmogorov Equation (without Advection)

[ ]( , ) ( , )P x t P x tt x

β α

β α

∂ ∂= Β

∂ ∂

1

1 (1 )( , ) s i n(1 ) 2

tP x tx

β

α

α π απ β

−

+

Γ + ⎡ ⎤≈ ⎢ ⎥Γ + ⎣ ⎦

Has Levy stable asymptotic power-lawevolution

with 1Β =

Fokker Plank Kolmogorov Equation & Fractional FPKE

2

41( , )4

x

P x e ττπ τ

⎛ ⎞−⎜ ⎟⎜ ⎟Β⎝ ⎠=

Β1

1 (1 )( , ) s in(1 ) 2

P xx

β

α

τ α πατπ β

−

+

Γ + ⎡ ⎤≈ ⎢ ⎥Γ + ⎣ ⎦

x

( )P x

The Dirac’s delta at time zero plumes out at any time and evolves. In case of integer order FPKEthe shape is dome shaped, where as for fractional order FPKE the cusp shape evolves. Well at veryinitial stage say at time , in integer order cases there exists a large quantity at fairly large distance from origin-stating that the phase velocity requirement is very very large-infinite.

This is due to ideal Fickian case, was modified in 1948 by Catteneo, by having relaxation time constantin the Fick’s second law, and thus obtaining telegrapher’s equation giving finite phase velocity.

τ

0.001sτ =

Power law relation & Asymptotic Fractals and Scale invariance:

Often one encounters functional relation in the frequency domain of the form:for example . Such relations back transformed to

time-space by generalized differentiation

Fractional expressions can arise quite naturally from the functional relationsobserved in so called “Spectral Domain”.

Typical asymptotic fractal is of the form.

Scale invariance in power law states that there is no small scale. Let then. A scaling in time axis results in simple scaling of ordinate.

The law is not altered by scale change. In log-log plot this means shifton axis and shift in the ordinate.

This scale invariance however is the cause of stability of the systems followingpower laws.

( )Fαω ω− exp( )αω ωτ− −( ) ( )F D f tα αω ω− −↔

*( ) ( , / )tG t tβ

γ β ττ

−⎛ ⎞≈ ⎜ ⎟⎝ ⎠

( )y t t β=

( )( ) ( )y t t y tβ βλ λ λ= =

( )y t t β=

lo g λ l o g t lo gβ λ

Fractional Kinetics FFPK with Fractal Support (in Chaotic Dynamics)

It is recently observed that numerous physical processes do not satisfy principle of short timememory randomness-and that long power like tails in distributions of different time-scales can frequently occur in physical systems.

For the cases when Fractional Kinetics can be applied there is ‘fractal’ support in ‘phase-space’ description of system. The origin of fractal support comes from strong non-uniformity of the ‘phase-space’ where special zones of ‘particle trapping’ exists.

The (quasi) traps can have fractal structure and so can the set of ‘time-intervals’ that trajectory spends in the trap. This is how we loose universality of the kinetics description of a system (with chaotic dynamics); because it depends on the ‘local space-time’ properties of the support (or more generally speaking on the phase-space topology).

Meaning in the Fractal Support system does time flow uniformly, with unit speed? As the aberration can be in space that may evolve as temporal non-uniformity!!

Law of irreversibility and Generalizing the Classical Calculus & Dynamics

Reversibility is to idealize. Its validity or applicability in physical experiments and processesdepends on the degree to which system can be isolated (or decoupled) from past historyand its environment.

Classical dynamic evolution problem is we generalize the same to:( ) ( )d f t f tdt

= Β

( ) ( )d f t f td t

α

α = Β B is operator on ‘phase space’, Banach space .Banach spaces, including spaces of continuous functions spaces etc.

2 22

2 2

g gct x

∂ ∂=

∂ ∂can be recast to ( ) ( )d f t f t

d t= Β with g

fh

⎛ ⎞= ⎜ ⎟

⎝ ⎠Even

and0 1

.1 0

cx

⎛ ⎞ ∂Β = ⎜ ⎟ ∂⎝ ⎠ d

d tWhere represents basic symmetry & basic principle oflocality. The fractional derivative whereas says about the historymemory, heredity and principle of non-locality. Time evolution isirreversible. Processes are having history (Non-Markovian). Let us postulatethat time evolution is irreversible and reversibility is idealism!!

/d dtα α

Time evolution:1

0 0

( ) ( )( ) ( )( ) lim lim t

t t

E f s f sd f s f s tf sd s t t

−

→ →

−− −= = −

dd tGives as infinitesimal generator of time translation defined by 1 ( ) ( ) ( )tE t f s f s t− = −

( ) ( )d f t f tdt

α

α = ΒWhereas abandons as generator of time evolution, represented by convolution with ‘memory-kernel’ and thus is replaced by:

1 ( )tE t−

10 0

0

( ) ( ) ( ) ( )t ts d sE t f t E s f t Kt t

αα

∞− − ⎛ ⎞= ⎜ ⎟

⎝ ⎠∫

( 1)sK xtα δ⎛ ⎞ = −⎜ ⎟

⎝ ⎠While with memory Kernel then the translation is 1

t tE Eα− −=

A simple translation with unit ‘speed’ reflects the idea of time ‘flowing’ uniformlywith constant velocity. The idea is embodied in measuring time by comparisonwith periodic clocks (processes). A competing idea related to ‘flow of time’represented by the convolution

is to measure time with non-periodic clocks such as decay or aging process!!

( )10 0

0

( ) ( ) ( ) ( )t tE t f t E s f t K u d uαα

∞− −= ∫

Irregularity & its identification

Normal Condition Beats: 90/70/90/70/90/70/90/70/90/70/90/70/90/70/90/70Abnormal Condition Beats:90/70/70/90/90/90/70/70/90/90/70/90/70/70/90/70

These two series have same mean, median, and variance and the two values (90 or 70) have the same probability of occurring ½ . Statistics fail to distinguish this; yet they are different. In first, one finds the next outcome with absolute certainty. In the second series we only know that next outcome will be either 90 or 70, but our guess will be wrong in 50% cases.

Fractals and multi-fractal functions and corresponding curves or surfaces are found in numerous places in non-linear and non-equilibrium phenomena. Like turbulence, Brownian paths, attractors of some dynamics, economics, seismology records are for examples of occurrence of continuous but highly irregular (non-differentiable) curves and surfaces.

Random functions have defined mean and variance (standard deviation). The non-random (power-law) functions have non definite mean or variance. Graphs are fractal set (self-similar) with no smallest scale.

Connection between ‘local’ scaling behavior (fractal dimension) and ‘Order of Fractional Derivative’ is interesting to ‘instrument’ the irregularity of non-linear dynamic systems.

Irregular Rough Functions with fractal dimension to identify

Uni-fractals have same fractal dimensions whereas the multi-fractals have several fractal dimensions.

Fractal dimension is measure of roughness or irregularity or chaotic curves, irregular curves non-differentiable curves.

0

( ) co s( )

30 1, 0 , 12

lo glo g 2

n n

n

F

f x a b n

a b a b

adb

π

π

∞

=

=

< < > > +

=+

∑Weistrauss

T FE

d d E

C R

< <

∈1 2Fd< <

Mandelbrot-Richardson Plot:

Plot of Logarithm of Length and Log of step size is Mandelbrot-Richardson Plot. The slope of that curve is S then fractal dimension is (1-S) . The S of the curve is equal to or less than zero.

log ( )L r

lo g r

0

lo g ( )1 limlo gF r

L rdr→

= −

lo g r

log ( )L rA B

CD

If the calculations of the length of the curve is performed with a step size that is too long, the main structure of the line described which gives a flat section (D), when the step size is too small, much less than thesampling interval, we are not able to recognize any new structure in the curve again results in flat section(A). The region (B) and (C) indicate two uncorrelated parts in the (irregular rough) signal.

Roughness Exponent:

For a function if there exists a polynomial of degree and a constant such that

The supremum of all exponents is Holder Exponent, whichcharacterizes singularity strength. Holder Exponent describes the ‘local’ regularity (roughness) of function

at

Higher the exponent, more regular is local behavior. Holder exponent is local and is related to Hurst Exponent as Where is average holder exponent in the defined interval. To findLocal Holder exponent one needs Wave-Let Transforms to characterize singularity

( )f x ( )nP x n h<C

0 0( ) ( ) hnf x P x x C x x− − ≤ −

0( ) ( , 1 )h x n n∈ +

( )f x 0x x=

( )1

2i iY tΔ = Δ is ‘square-root’ scaling of Brownian Motion, with as Hurst exponent . To

find Hurst exponent one needs Rescaling (R/S) method of time series0 .5H =

t

( )Y t

Roughness of Graph is: lo g / lo gi iH Y t= Δ Δ

1h H= − h

Holder Exponent & (box) dimension of graph (function)

r

r

1.00

Hr≤

r Intervals

( )f x

x

We cover the shape with boxes of size and find how many boxesrequired to cover the shape. Then

r( )N r 0

lo g ( )limlo g (1 / )B r

N rdr→

=

1. The function defined in2. Divide into equal intervals or almost equal intervals.3. Above each interval make column of width .4. In the situation of scaling condition with Holder Exponent as: means in each of the column of graph of passes through a height5. So the number of boxes needed to cover the part of the graph in that column is about (height of graph) (height of box)=6. The number of columns is in the length 1.00.7. The number of these boxes of side needed to cover the entire graph is

8.

( )f x0 1x≤ ≤ r

r( ) Hy xΔ = Δ

( )f x Hr

÷ 1H Hr r r −÷ =1 / r

r1 2( ) (1 / ) ( )H Hr r r N r− −× = =

2

0

l o g ( ) l o gl i m 2l o g (1 / ) l o g (1 / )

H

B r

N r rd Hr r

−

→= = = −

Box-dimension of irregular graph and Holder’s exponent

The Holder exponent of the Brownian path is ½ , the scaling; hence the box, or the fractal dimension of Brownian motion is 1.5. The white noise is also scaled by ½ and has box or fractal dimensions 1.5.

[ ]0 , 1

Equivalence of Mandelbrot-Richardson fractal dimension and the Box-dimension

1 1

1

l o g ( ) l o g ( )1 1l o g l o g ( 1 / )

l o g ( )1l o g ( 1 / )

( ) ( 1 / )l o g ( )l o g ( 1 / )

( )( ) ( ) ( 1 / )

( )

F F

B

B B

F

F

d d

B

d

d d

B F

L r L rdr r

L rdr

L r r rN rd

rN r rN r L r rL r r r rd d

− −

−

− −

= − = +

− =

= =

=

== ×

= × ==

Both the methods give fractal dimension of irregular graphs called fractal dimension or Hausdroff’sdimension.

The efficient method of estimation of irregularity is good R&D.

Several dimensions an overview:

1. Embedding Euclidian dimension: The regular dimension which holds the structure.

1. Fractal dimension : Describes irregularity and roughness, locally or globally. For self similar the dilation symmetry is uniform, but the self affine has anisotropic dilation symmetry.

2. Spectral dimension: Described by density of states or the relaxation of stimulus and order of it at asymptotic late times.

d

Fd

Sd

F Sd d d≥ ≥

Scaling:

( ) ( )( )

q qq

q q

d f x d f xd x d x

λ λλλ

=

This property makes suitable for study of scaling. Local scaling behavior is itrelated to order of fractional derivative? However, this scaling property of fractional derivative implies study of ‘self-similar’ objects, distributions etc.Let then . A scaling in the time axis resultsin simple scaling of ordinate. The law is not altered. In log-log plotthis means a shift of on the axis and shift in the ordinate. This scale-invariance is the cause of stability of system followingpower-law. Power-law relaxation is observed in various physical systems. Thusone often encounters algebraic relaxation withTherefore one often encounters fractional relations in frequency domain

for example . Fractional relationsof this self-similar (fractal) form can arise quite naturally in spectral domain.in time domain fractal form is ; this form has late time power decay law.

( )y t t β= ( ) ( ) ( )y t t y tβ βλ λ λ= =( )y t t β=

l o g λ l o g t l o gβ λ

( )t t αφ −≈ 0 1α< <

( )Fαω ω− e x p ( )αω ω τ− −

*( ) ( / ) ( , / )g t t tατ γ β τ−≈

Fractal Dimensions of self-similar figures0

loglim1log

FNd

ε

ε→

=⎛ ⎞⎜ ⎟⎝ ⎠

1Fd =2Fd =

2N =

3N =

2, 1 / 2r ε= =

3, 1/3r ε= =4

2 , 1 / 2Nr ε

== =

93 , 1 / 3

Nr ε

== =

32, 1 / 2

log 3 log 3 / log 2 1.5851log 12

F

Nr

d

ε== =

= = =⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

‘Box’ could be a circle of radius ‘r’ or square of side ‘r’ a sphere of radius ‘r’or triangle of side ‘r’ or segment of side ‘r’. Infinitesimally all are same!!

There is relation of fractality and fractional calculus, thus we have introduced theconcept of fractals and fractal dimensions.

A non-differentiable curve is Fractionally differentiable (locally).

Thus a physical process which makes a Phase change where differentiability is lost, fractional derivative (locally) gives the idea of the enlargement/manifestation of the physics of phase change.

Fractal Dimensions Fractality & Fractional calculus-a passing mention

Let us enter the World of Fractional Calculus